The eXtended Finite Element Method - Code Aster - Code_Aster and Salome-Meca course material GNU FDL...
Transcript of The eXtended Finite Element Method - Code Aster - Code_Aster and Salome-Meca course material GNU FDL...
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The eXtended Finite Element Method
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Outline
Motivation
Theoretical aspects
Level Sets Method
X-FEM enrichments
Application to code_aster
Crack definition (Level Sets)
Enriched Finite Elements (X-FEM)
Post-processing in fracture mechanics
Visualisation
Advanced tools
Mesh refinement
Crack propagation
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Outline
Motivation
Theoretical aspects
Level Sets Method
X-FEM enrichments
Application to code_aster
Crack definition (Level Sets)
Enriched Finite Elements (X-FEM)
Post-processing in fracture mechanics
Visualisation
Advanced tools
Mesh refinement
Crack propagation
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Goals
Study of cracked structures without explicitly meshing crack
Automatic crack propagation on a single mesh
Finite Element Method (FEM)
Crack is explicitly meshed
A long time (human intervention) is needed to mesh complex
structures
Re-meshing is required if changing the crack geometry
(parametric study) or position (propagation)
eXtended Finite Element Method (X-FEM)
Simple mesh (does not respect the crack geometry)
Unique mesh for entire propagation process
Simple extension of FEM (relatively suitable for
implementation in a EF software)
Motivation
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Outline
Motivation
Theoretical aspects
Level Sets Method
X-FEM enrichments
Application to code_aster
Crack definition (Level Sets)
Enriched Finite Elements (X-FEM)
Post-processing in fracture mechanics
Visualisation
Advanced tools
Mesh refinement
Crack propagation
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Representation of interface by level set
Main idea: interface = iso-zero of a “distance function”
In mechanics: used for modelling interfaces between materials,
edges, voids, cracks
LS > 0
LS < 0
LS = 0
Iso-values of LS for a spherical
interface
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Crack
plane
(LSN = 0)
Crack(LSN = 0 LST < 0)
xLSN(x)
LST(x)
Iso-values of LST
Iso-zero of LSN
Representation of crack by level sets
2 level sets
are needed
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X-FEM enrichment
Finite elements enriched with
Heaviside function
Finite elements enriched with
asymptotic functions (~r)
X-FEM = Two enrichmentsEnrichment with a Heaviside function to represent a discontinuous field (displacement
jump) across the interfaceNB : The actual formulation is more complex but not detailed here for the sake of simplicity
Enrichment with asymptotic functions near the crack tip
( ) ( ) ( )i i
i i
N N H i iu x a x x xb 1,4 ,i
i
N B r 1,4
ixc
Classical part
(continuous)
Heaviside
enrichment
(discontinuous)
Asymptotic
enrichment
(singular)
sin2
cos,sin2
sin,2
cos,2
sin rrrrB
where
lst
lsnlstlsnr 122 tan, (signed distance functions)
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Outline
Motivation
Theoretical aspects
Level Sets Method
X-FEM enrichments
Application to code_aster
Crack definition (Level Sets)
Enriched Finite Elements (X-FEM)
Post-processing in fracture mechanics
Visualisation
Advanced tools
Mesh refinement
Crack propagation
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Crack Definition (Level sets)
Definition of a crack (or an interface) with the command DEFI_FISS_XFEM
Choice of the mesh
Choice of a crack or a interface
Definition of the crack or the interface with level sets
Definition of the enrichment zone (optional)
Definition of the type of enrichment (optional)
Mesh
Finite Element
Enrichment
Computation
Resolution
Post-processing
SIF,
propagation
Visualisation
Crack definition
(level sets)
« sane » mesh
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Crack Definition (Level sets)
Choice of a mesh
FISS=DEFI_FISS_XFEM(MAILLAGE=MAILLA,
…
)
The sane mesh is provided via the keyword MAILLAGE (mandatory)
Level sets will be computed at all the nodes of this mesh
Mesh
Finite Element
Enrichment
Computation
Resolution
Post-processing
SIF,
propagation
Visualisation
Crack definition
(level sets)
« sane » mesh
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Crack Definition (Level sets)
A crack An interface
Choice of a crack (fissure in French) or an interface
FISS=DEFI_FISS_XFEM(MAILLAGE=MAILLA,
TYPE_DISCONTINUITE = / 'FISSURE', [DEFAULT]
/ 'INTERFACE'
)
Mesh
Finite Element
Enrichment
Computation
Resolution
Post-processing
SIF,
propagation
Visualisation
Crack definition
(level sets)
« sane » mesh
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Crack Definition (Level sets)
Choice of a crack (fissure in French) or an interface
FISS=DEFI_FISS_XFEM(MAILLAGE=MAILLA,
TYPE_DISCONTINUITE = / 'FISSURE',
/ 'INTERFACE'
DEFI_FISS=_F()
)
Level sets defined under the keyword DEFI_FISS
4 available methods:
Analytical functions
Pre-defined geometries
Projection onto group of elements
Lecture of pre-computed LSN and LST fields
Mesh
Finite Element
Enrichment
Computation
Resolution
Post-processing
SIF,
propagation
Visualisation
Crack definition
(level sets)
« sane » mesh
(only this method is
employed in this workshop)
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x
zy
H
a
Crack Definition (Level sets)
Method 1: analytical functions
LSN=FORMULE(NOM_PARA=('X','Y','Z'),VALE='Z-H')
LST=FORMULE(NOM_PARA=('X','Y','Z'),VALE=‘Y-a')
FISS=DEFI_FISS_XFEM(MAILLAGE=MAILLA,
DEFI_FISS=_F( FONC_LN=LSN,
FONC_LT=LST,),
…
)
Formula of LSN and LST are available only for very simple crack
geometries
Rapidity
Formula of LSN and LST not available for
complex crack geometries
Mesh
Finite Element
Enrichment
Computation
Resolution
Post-processing
SIF,
propagation
Visualisation
Crack definition
(level sets)
« sane » mesh
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Crack Definition (Level sets)
Method 2: pre-defined geometries
FORM_FISS = SIMP (
# type of cracks :
3d : "ELLIPSE","CYLINDRE","DEMI_PLAN" (half-plane),“RECTANGLE“
Mesh
Finite Element
Enrichment
Computation
Resolution
Post-processing
SIF,
propagation
Visualisation
Crack definition
(level sets)
« sane » mesh
2d : "DEMI_DROITE","SEGMENT","ENTAILLE"
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Crack Definition (Level sets)
Method 2: pre-defined geometries
FISS=DEFI_FISS_XFEM(MAILLAGE=MAILLA,
DEFI_FISS=_F( FORM_FISS='ELLIPSE',
DEMI-GRAND_AXE = 0.4,
DEMI-PETIT_AXE = 0.2,
CENTRE = (0.5, 0.5, 0.5),
VECT_X = (0. , 1. , 0.2 ),
VECT_Y = (-1., 0. , 0.4 ),
))
This method is highly recommended
If a crack geometry is not available in the catalogue, propose a new
development to the code_aster team
Low risk of user error
Rapidity
Automatic crack front
orientation
Mesh
Finite Element
Enrichment
Computation
Resolution
Post-processing
SIF,
propagation
Visualisation
Crack definition
(level sets)
« sane » mesh
Half major axis
Half minor axis
Direction of major axis
Direction of minor axis
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Crack Definition (Level sets)
x
zy
H
a
Groups of 2D elements : MA_FISS
Groups of 1D elements : MA_FOND
Capability of building complex cracks
Time consuming
Method 3: projection onto groups of elements
FISS=DEFI_FISS_XFEM(MAILLAGE=MAILLA,
DEFI_FISS=_F( GROUP_MA_FISS=‘MA_FISS’,
GROUP_MA_FOND=‘MA_FOND’),
…)
Mesh
Finite Element
Enrichment
Computation
Resolution
Post-processing
SIF,
propagation
Visualisation
Crack definition
(level sets)
« sane » mesh
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Crack Definition (Level sets)
Method 4: lecture of a pre-computed LSN and LST
fields
TBLSN=LIRE_TABLE(UNITE=38,
FORMAT='ASTER',
NUME_TABLE=1,
SEPARATEUR=' ',
TITRE='level set normale',)
CHLSN=CREA_CHAMP(OPERATION='EXTR',TABLE=TBLSN,
TYPE_CHAM='NOEU_NEUT_R',MAILLAGE=MAILLAG1)
…
FISS=DEFI_FISS_XFEM(MAILLAGE=MAILLA,
DEFI_FISS=_F( CHAM_NO_LSN=CHLSN,
CHAM_NO_LST=CHLST),
…)
Mesh
Finite Element
Enrichment
Computation
Resolution
Post-processing
SIF,
propagation
Visualisation
Crack definition
(level sets)
« sane » mesh
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Crack Definition (Level sets)
optional keyword
Definition of enrichment zoneRestriction of the crack definition zone with GROUP_MA_ENRI
FISS=DEFI_FISS_XFEM(MAILLAGE=MAILLA,
DEFI_FISS=_F(GROUP_MA_FISS=‘MA_FISS’,
GROUP_MA_FOND=‘MA_FOND’),
GROUP_MA_ENRI = ‘M_LEFT’,
…)
advanced
Mesh
Finite Element
Enrichment
Computation
Resolution
Post-processing
SIF,
propagation
Visualisation
Crack definition
(level sets)
« sane » mesh
crackCrack extension
(un-intentional)
Extension of the crack
front
‘M_LEFT’, ‘M_RIGHT’,
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Crack Definition (Level sets)
RAYON_ENRI / NB_COUCHES
TOPOLOGIQUE GEOMETRIQUE
Definition of the type of enrichment
FISS=DEFI_FISS_XFEM(MAILLAGE=MAILLA,
DEFI_FISS=_F(GROUP_MA_FISS=‘MA_FISS’,
GROUP_MA_FOND=‘MA_FOND’),
GROUP_MA_ENRI = ‘M_VOL’,
TYPE_ENRI_FOND = / 'TOPOLOGIQUE' [DEFAULT]
/ 'GEOMETRIQUE'
optional keyword
advanced
Mesh
Finite Element
Enrichment
Computation
Resolution
Post-processing
SIF,
propagation
Visualisation
Crack definition
(level sets)
« sane » mesh
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Maillage
Enrichissement
des éléments
finis
Calcul -
Résolution
Post-traitement
FICs, critère
propagation
Visualisation
Définition de la
fissure (level sets)
# Crack definitionFISS = DEFI_FISS_XFEM(MAILLAGE = MAILLA,
...
DEFI_FISS=_F(...),)
# Sane model definitionMOD_SAIN = AFFE_MODELE(MAILLAGE = MAILLA,
...
AFFE = _F(...),)
Enriched Finite Elements (X-FEM)
Mesh
Finite Element
Enrichment
Computation
Resolution
Post-processing
SIF,
propagation
Visualisation
Crack definition
(level sets)
Available modelisations: 3D, C_PLAN, D_PLAN, AXIS
Linear or Quadratic elements
The crack and the sane model must be defined on the same mesh
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Maillage
Enrichissement
des éléments
finis
Calcul -
Résolution
Post-traitement
FICs, critère
propagation
Visualisation
Définition de la
fissure (level sets)
# Crack definitionFISS = DEFI_FISS_XFEM(MAILLAGE = MAILLA,
...
DEFI_FISS=_F(...),)
# Sane model definitionMOD_SAIN = AFFE_MODELE(MAILLAGE = MAILLA,
...
AFFE = _F(...),)
# Creation of a model with enriched elements (X-FEM)MOD_FISS=MODI_MODELE_XFEM(MODELE_IN = MOD_SAIN,
FISSURE = FISS)
Enriched Finite Elements (X-FEM)
MOD_FISS must be used from now on (loadings, material…)
MOD_SAIN is the sane model
FISS is the crack
FISSURE accepts also a list
of cracks / interfaces
Mesh
Finite Element
Enrichment
Computation
Resolution
Post-processing
SIF,
propagation
Visualisation
Crack definition
(level sets)
Computation for non-linear studies in quasi-static : STAT_NON_LINE, MECA_STATIQUE,
THER_LINEAIRE
RESU=STAT_NON_LINE(CHAM_MATER=CHAMPMAT,
EXCIT=(_F(CHARGE=CH1,),),
MODELE=MOD_FISS,
…
SOLVEUR=_F(METHODE='MUMPS'))
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No new keyword:
The crack is known
via the model
(enrich elements)
if contact on crack faces: MUMPS
solver
Resolution
Maillage
Enrichissement
des éléments
finis
Calcul -
Résolution
Post-traitement
FICs, critère
propagation
Visualisation
Définition de la
fissure (level sets)
Mesh
Finite Element
Enrichment
Computation
Resolution
Post-processing
SIF,
propagation
Visualisation
Crack definition
(level sets)
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Resolution
Focus on the sub-element cutting processContext: numerical integration of rigidity terms and right-hand side force terms for
elements cut by the crack
On an element cut through by the crack, the integrand might be discontinuous
Numerical integration with Gauss method on the whole element is not possible
Solution: sub-element cutting (only for integration purpose)
elementssub sese
t
e dBDBK
with e
e
t
e dBDBK 4,1,
FFHB kkji
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Post-processing in fracture mechanics
PK=POST_K1_K2_K3( MODELISATION='3D',
MATER=ACIER,
FISSURE = FISS,
RESULTAT = UTOT1,
NB_POINT_FOND = 15,
ABSC_CURV_MAXI = 0.3)
TABFIC=CALC_G(RESULTAT=RESU,
OPTION='CALC_K_G',
THETA=_F(FISSURE=FISS,
R_INF=0.1,
R_SUP=0.3,
NB_POINT_FOND = 15))
Limit the post-
processing on few
points along the crack
front
reduction of the
oscillations
Idem for meshed cracks (FEM)POST_K1_K2_K3
CALC_G
Maillage
Enrichissement
des éléments
finis
Calcul -
Résolution
Post-traitement
FICs, critère
propagation
Visualisation
Définition de la
fissure (level sets)
Mesh
Finite Element
Enrichment
Computation
Resolution
Post-processing
SIF,
propagation
Visualisation
Crack definition
(level sets)
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Post- processing for visualisation
Impossible to visualise the crack opening on the computational
mesh (sane mesh)
Need for a visualisation tool
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MA_VISU = POST_MAIL_XFEM(MODELE = MOD_FISS)
MO_VISU = AFFE_MODELE(MAILLAGE = MA_VISU, AFFE=_F(…))
RES_VISU = POST_CHAM_XFEM(MODELE_VISU = MO_VISU,
RESULTAT = RESU)
IMPR_RESU(RESU=_F(RESULTAT = RES_VISU))
Post- processing for visualisation
Creation of a mesh for visualisation purpose
Creation of result fields on this mesh
Maillage
Enrichissement
des éléments
finis
Calcul -
Résolution
Post-traitement
FICs, critère
propagation
Visualisation
Définition de la
fissure (level sets)
Mesh
Finite Elements
Enrichment
Computation
Resolution
Post-processing
SIF,
propagation
Visualisation
Crack definition
(level sets)
28 - Code_Aster and Salome-Meca course material GNU FDL Licence
Outline
Motivation
Theoretical aspects
Level Sets Method
X-FEM enrichments
Application to code_aster
Crack definition (Level Sets)
Enriched Finite Elements (X-FEM)
Post-processing in fracture mechanics
Visualisation
Advanced tools
Mesh refinement
Crack propagation
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Mesh refinement
von Mises stresses near crack tip
r
x
y
Stress is proportional to 1/r, with r the distance to the
crack front
For both crack representations using FEM or X-FEM, mesh
needs to be fine enough for accurate SIF computation
Importance of the mesh refinement near the crack front
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Mesh refinement
A “sane” mesh needs to be refined before launching the FE
analysis (adaptation a priori)
Refinement near the crack tip (2D) or near the crack front (3D)
Automatic operation using the distance to the crack front (computed by level sets)
Only elements nearest to the crack from are refined
X-FE analysisInitial
“sane”
meshLevel sets
Distance to
the crack-front
Refinement
software
(HOMARD)
Refined “sane” mesh
Automatic mesh refinement loop
SIFs
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Mesh refinement in Code_Aster
Refinement parameters (python)
Level sets computation on the initial mesh (DEFI_FISS_XFEM)
Computation of the refinement criteria (RAFF_XFEM)
Mesh refinement (MACR_ADAP_MAIL)
Parameters (python)
crack = DEFI_FISS_XFEM(…)
error = RAFF_XFEM(…)
refmesh =
MACR_ADAP_MAIL(…)
mesh = LIRE_MAILLAGE()
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Mesh refinement in Code_Aster
Refinement parametersh0: mesh size in the area of interest
hC: mesh size after refinement → hC ≈ a/20 (a: crack’s
depth)
Calculation of the refinement number
𝑛𝑏𝑟𝑎𝑓𝑓 = 𝐸 𝑛 𝑤𝑖𝑡ℎ 𝑛 =ln ℎ0 − ln(ℎ𝐶)
ln(2)
Real number of iterations for adaption loop
𝑛𝑏𝑖𝑡𝑒𝑟 = 𝑛𝑏𝑟𝑎𝑓𝑓 + 1
Calculation of real mesh size after refinement(s)
ℎ =ℎ0
2𝑛𝑏𝑟𝑎𝑓𝑓
Parameters (python)
crack = DEFI_FISS_XFEM(…)
error = RAFF_XFEM(…)
refmesh =
MACR_ADAP_MAIL(…)
mesh = LIRE_MAILLAGE()
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Mesh refinement in Code_Aster
Level sets computation on the initial mesh (DEFI_FISS_XFEM)
FISS=DEFI_FISS_XFEM(MODELE=MO,
DEFI_FISS=_F(FORM_FISS = 'ELLIPSE',
DEMI_GRAND_AXE = 2.,
DEMI_PETIT_AXE = 2.,
CENTRE = (0.,0.,0.),
VECT_X = (1.,0.,0.),
VECT_Y = (0.,1.,0.),),
GROUP_MA_ENRI='CUBE')
Parameters (python)
crack = DEFI_FISS_XFEM(…)
error = RAFF_XFEM(…)
refmesh =
MACR_ADAP_MAIL(…)
mesh = LIRE_MAILLAGE()
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Mesh refinement in Code_Aster
Computation of the refinement criteria (RAFF_XFEM)
CH_ERR=RAFF_XFEM(TYPE = / ‘DISTANCE’ [DEFAULT]
/ ‘ZONE’, [RECOMMENDED]
FISSURE = FISS,
RAYON = R, [IF TYPE=‘ZONE’]);
Goal: to localize the area near the crack front
Type = ‘DISTANCE’
Nodes field
r = distance to the crack front
Near crack front, error 0
Far crack front, error -
Type = ‘ZONE’
Elements field
1 in the refinement area (radius 𝑅 = 5ℎ (s.32))
0 elsewhere
r
lstlsnr
error
22
Parameters (python)
crack = DEFI_FISS_XFEM(…)
error = RAFF_XFEM(…)
refmesh =
MACR_ADAP_MAIL(…)
mesh = LIRE_MAILLAGE()
2D: disk around the crack type
3D: tore around the crack type
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Mesh refinement in Code_Aster
Refinement : Homard (MACR_ADAP_MAIL)
RAFF_XFEM → Type = ‘DISTANCE’
MACR_ADAP_MAIL(ADAPTATION='RAFFINEMENT',
CHAM_GD = CH_ERR,
CRIT_RAFF_PE = 0.1,
USAGE_CMP = ‘RELATIF’,
MAILLAGE_N = MA,
MAILLAGE_NP1 = CO(‘MA2’),)
RAFF_XFEM → Type = ‘ZONE’
MACR_ADAP_MAIL(ADAPTATION='RAFFINEMENT',
CHAM_GD = CH_ERR,
CRIT_RAFF_ABS = 0.5,
DIAM_MIN = hC,
MAILLAGE_N = MA,
MAILLAGE_NP1 = CO(‘MA2’),)
Parameters (python)
crack = DEFI_FISS_XFEM(…)
error = RAFF_XFEM(…)
refmesh =
MACR_ADAP_MAIL(…)
mesh = LIRE_MAILLAGE()
Percentage of
elements to be
refined
Refinement of the
element with the
largest error
All elements
with a value
greater than 0.5
will be refined
To avoid refining
an element that
already reaches
the target size
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Mesh refinement in Code_Aster
MA=LIRE_MAILLAGE()
MO=AFFE_MODELE(MAILLAGE=MA,
AFFE=_F(TOUT=‘OUI’,PHENOMENE='MECANIQUE',
MODELISATION='3D'))
# crack definitionFISS = DEFI_FISS_XFEM(MODELE=MO,
DEFI_FISS=_F(FORM_FISS = 'ELLIPSE',
DEMI_GRAND_AXE = 2.,
DEMI_PETIT_AXE = 2.,
CENTRE = (0.,0.,0.),
VECT_X = (1.,0.,0.),
VECT_Y = (0.,1.,0.),),
GROUP_MA_ENRI='CUBE')
# error fieldCH_ERR=RAFF_XFEM(FISSURE=FISS)
# Mesh refinement (call of HOMARD software)
MACR_ADAP_MAIL(ADAPTATION='RAFFINEMENT',
CHAM_GD = CH_ERR,
CRIT_RAFF_PE = 0.1,
TYPE_VALEUR_INDICA='V_RELATIVE',
NOM_CMP_INDICA = 'X1',
MAILLAGE_N = MA,
MAILLAGE_NP1 = CO(‘MA2’),)
Refined mesh
Possibility to
use python for
a loop
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Examples of mesh refinement
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Fatigue Propagation (constant amplitude)
Propagation of a 2d crack in a plate under tension with a hole
Computation of fatigue propagation: 3 ingredients
A propagation law (gives the increment of crack advance)
A bifurcation criterion (gives the bifurcation angle)
An algorithm for the crack update
common to meshed
and non-meshed
cracks
Specific to X-FEM +
level sets framework
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Fatigue Propagation (constant amplitude)
Ingredient n°1: propagation law
Paris’ law
C and m : material parameters
Equivalent stress intensity factor
Propagation driven by a maximal advance increment : amax
Ingredient n°2: bifurcation criterion
“Maximum Hoop Stress” (Erdogan, Sih)
8
4
1arctan2
2
II
III
II
I
K
KKsign
K
K
2
2
222
1
1IIIIIIeq KKKK
meqKCdN
da .
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Fatigue Propagation (constant amplitude)
Ingredient n°3: crack update
Crack update = level sets update (X-FEM + level set framework)
3 methods available in Code_Aster for the level sets update (operator PROPA_FISS)
« naïve » approach: MAILLAGE method
SIMPLEX method
UPWIND method
GEOMETRIQUE method
Numerical methods for
level set update
Ingredient n°3: crack update« naïve » approach: MAILLAGE method
UPWIND method
Auxiliary regular grid is required for complex geometry
SIMPLEX method
No auxiliary regular grid
Robust method
GEOMETIQUE method
Faster than the other techniques and capable of modeling 3D non-planar crack growth.
Initial crack
bifurcation angle:
Advance increment: da
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Fatigue Propagation (constant amplitude)
level sets update = adding a segment/an elementary
surface (due to crack advancement) to the crack
discretization, then computation of level sets by
orthogonal projection of the nodes on this new crack
location
Already in DEFI_FISS_XFEM (projection onto groups of
elements)
More information
R7.02.12
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Fatigue Propagation (constant amplitude)
Propagation of a quarter-disk crack with
automatic mesh adaptation
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Conclusion on X-FEM in code_aster (V12)
Behaviour law
linear elasticity, non-linear, elasto-plasticity
Modelisations
plane strain, plane stress, axi-symmetrical, 3D
Loadings
pressure on crack faces; displacement boundary conditions near the
crack (symmetry conditions); rotation, gravity
Contact on crack faces
Finite elements
linear or quadratic
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Conclusion on X-FEM in code_aster (V12)
One or multiple crack, crack net, crack branching
Post-processing in fracture mechanics:
CALC_G and POST_K1_K2_K3
Visualisation
displacements and stresses on a “cracked” mesh
Crack propagation (2d, 3d in-plane and out-of-plane)
Modal analysis
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References
Propagation and crack
branching
General user documentation[U2.05.02] how to use X-FEM
Operator documentation[U4.82.08] Operator DEFI_FISS_XFEM
[U4.41.11] Operator MODI_MODELE_XFEM
[U4.82.11] Operator PROPA_FISS
[U4.82.21] Operator POST_MAIL_XFEM
[U4.82.22] Operator POST_CHAM_XFEM
General user documentation[R7.02.12] eXtended Finite Element Method
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End of presentation
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Appendix for TP forma05c
X-FEMMesh
Finite Element
Enrichment
Computation
Resolution
Post-processing
SIF,
propagation
Visualisation
Crack definition
(level sets)
« sane » mesh
DEFI_FISS_XFEMOne by crack
IDEM FEM
MODI_MODELE_XFEM
CALC_G, POST_K1_K2_K3
POST_MAIL_XFEM, POST_CHAM_XFEM